Complexity

Volume 2017, Article ID 9419024, 14 pages

https://doi.org/10.1155/2017/9419024

## Predictability of Extreme Waves in the Lorenz-96 Model Near Intermittency and Quasi-Periodicity

Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, Netherlands

Correspondence should be addressed to A. E. Sterk; ln.gur@krets.e.a

Received 14 May 2017; Revised 11 July 2017; Accepted 16 August 2017; Published 24 September 2017

Academic Editor: Davide Faranda

Copyright © 2017 A. E. Sterk and D. L. van Kekem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a method for quantifying the predictability of the event that the evolution of a deterministic dynamical system enters a specific subset of state space at a given lead time. The main idea is to study the distribution of finite-time growth rates of errors in initial conditions along the attractor of the system. The predictability of an event is measured by comparing error growth rates for initial conditions leading to that event with all possible growth rates. We illustrate the method by studying the predictability of extreme amplitudes of traveling waves in the Lorenz-96 model. Our numerical experiments show that the predictability of extremes is affected by several routes to chaos in a different way. In a scenario involving intermittency due to a periodic attractor disappearing through a saddle-node bifurcation we find that extremes become better predictable as the intensity of the event increases. However, in a similar intermittency scenario involving the disappearance of a 2-torus attractor we find that extremes are just as predictable as nonextremes. Finally, we study a scenario which involves a 3-torus attractor in which case the predictability of extremes depends nonmonotonically on the prediction lead time.

#### 1. Introduction

Classical extreme value statistics is concerned with the asymptotic distribution of large values in time series of random variables. The theory, which is based on the extreme value and generalized Pareto distributions, is well developed for stochastic processes both with and without serial dependence; see the text books [1–7]. A recent development is the application of extreme value statistics in the setting of* deterministic* dynamical systems. The main idea is to evaluate a scalar observable along the evolution of a system and to study under which conditions the same extreme value laws hold as in the case of stochastic processes. Geophysical applications, in which dynamical systems arise as models and observables are physical quantities like wind speed or temperature, form an important motivation for the development of the theory. Very recently, Lucarini et al. [8] published the first text book on extremes in dynamical systems which gives an excellent overview of the latest developments and also provides an extensive source of references.

Statistics only describe the behaviour of extremes over long periods of time. However, for the development of early-warning systems and risk mitigation strategies the short-term predictability of extremes is of great importance. This leads to the following question: how predictable are extremes? Bodai [9] summarizes three different conclusions that can be found in the literature:(1)Extremes are better predictable.(2)Extremes are less predictable.(3)Extremes can be better or less predictable depending on several factors.The first conclusion is supported by the work of Hallerberg et al. [10] who studied the predictability of extreme* increments* in first-order autoregressive process, wind speed recordings, and long-range correlated autoregressive moving averages. In all their examples extremes become better predictable with increasing event size. The results in [11] showed that in i.i.d. stochastic processes large increments are better predictable if the process is Gaussian, whereas large increments become less predictable if the underlying distribution has a power law tail. However, in the follow-up study [12], which is concerned with* threshold crossings* instead of* increments*, it was found again that extremes are always better predictable. The first conclusion is also supported by the work of Franzke [13, 14] in the context of dynamic-stochastic models. Bodai [9] argues that in dynamical systems stronger predictability of extremes may be typical but not universal. The third conclusion is supported by the work of Sterk et al. In [15] it was pointed out that the predictability of extreme values in dynamical systems depends on the observable, the attractor of the system, and the prediction lead time. In [16] it was shown how the tail of the distribution of wind speeds affects their predictability at high thresholds.

The predictability of extremes can be measured in different ways. By treating extreme events as binary events one can measure prediction skill by means of a receiver operator characteristic (ROC) curve which is a graph of the hit rate against the false alarm rate [9–14]. Another possible measure is the extreme dependency score developed by Stephenson et al. [17], which does not tend to zero for vanishingly rare events unlike scores such as the equitable threat score. Alternatively, when predictions are made using a dynamical model, predictability can be measured in terms of the growth rate of errors in the initial condition. The earliest studies on predictability in atmospheric models [18–20] computed the time needed for small errors in the initial condition to double in magnitude. This idea connects with traditional predictability measures for dynamical systems, such as Lyapunov exponents. The latter are asymptotic quantities that are computed for time tending to infinity, which also implies that they are independent of the initial condition [21]. Finite-time Lyapunov exponents and singular values measure the growth rate of errors over a finite time and typically they strongly depend on the initial condition and the prediction lead time. Measures of this type have been developed in celestial mechanics to separate chaotic from regular dynamics [22, 23], and they have been used to measure the growth of errors due to model perturbations [24] and the predictability of extremes [15].

Several papers demonstrated that finite-time error growth rates can show large fluctuations along the attractor of the system [25–31]. Benzi and Carnevale [32] argued that a ratio of the average growth rate to the most probable growth rate much smaller than 1 is an indication of enhanced predictability, which means that some events may be better predictable than others. A natural question is then what kind of dynamics can lead to enhanced predictability? For example, in dynamical systems with intermittency the dynamics switches between two or more different dynamical regimes and each regime can be associated with different predictability characteristics. The work in [33, 34] shows that in intermittent dynamical systems distributions of finite-time Lyapunov exponents are non-Gaussian and asymmetric and have heavy tails. Hence, in intermittent systems one can expect that some events are better predictable than others.

The aim of this paper is to demonstrate that the predictability of extremes depends on the dynamical regime of the model that is used for the predictions. In particular, we show that in weakly chaotic regimes of a dynamical system the predictability of extremes does not have universal properties. The main idea, which is in the spirit of [32], is to study the distribution of finite-time growth rates of errors in initial conditions along the attractor of the system. Comparing error growth rates of initial conditions leading to an event with all possible growth rates then gives a measure of the predictability of the event. We illustrate the method using the Lorenz-96 model [35]. On the one hand this model is simple enough for performing detailed numerical explorations. On the other hand the model has many dynamical features that are shared by a large class of geophysical models. The Lorenz-96 model can be interpreted as a model for traveling waves. The routes to chaos are myriad and different kinds of attractors can be found [36]. The bifurcation scenarios in the Lorenz-96 model can also be found in more complex geophysical models, such as the atmospheric and oceanic models studied in [37, 38]. We will focus in particular on predictability in the vicinity of bifurcations leading to intermittent and quasi-periodic dynamics.

The remainder of this paper is structured as follows. In Section 2 we explain how to quantify the predictability of an event in a general dynamical system. In Section 3 we introduce the Lorenz-96 model which we will use for our numerical experiments. For three values of the dimension of the model we investigate how the predictability of extreme waves in the model depends on intermittent or quasi-periodic nature of the dynamics. Section 4 concludes the paper with a summary and discussion of the results and suggestions for further research.

#### 2. Predictability of Dynamical Systems

This section explains the methodology of quantifying the predictability of an event in a dynamical system. In general, a deterministic dynamical system can be defined as a triple which consists of a state space , a time set , and an evolution operator , such that the following properties are satisfied:(i) is an additive half group: and for all also .(ii)For all and we haveWe also write . Particular examples that are included in this setting are discrete time systems, such as iterated maps, and continuous-time systems, such as flows of differential equations; see [39, 40]. In this paper we assume that the state space is a subset of the Euclidean space, but more generally can be a Riemannian manifold or a function space.

The predictability of a dynamical system is often quantified in terms of the growth rate of errors in the initial condition. Suppose that the initial condition is perturbed in the direction of ; then is the error growth rate over a time interval of length . Harle et al. [28] studied the statistics of these growth rates and their dependence on the parameters and in the setting of 2-dimensional dissipative and conservative maps. The error growth was found to increase exponentially fast with when is small. For larger values of the error growth follows a power law which depends on the magnitude of . In their paper it is suggested that these results are quite general.

In this paper we will make the idealized assumption that the initial perturbation size is infinitesimally small. Under this assumption the error at time is then given bywhere the derivative is taken with respect to the initial condition in the direction of the vector . The worst-case error growth over a time interval of length can be computed by maximizing the following Rayleigh quotient over all nonzero vectors :where denotes the Euclidean norm. A standard result in linear algebra [41] implies that the quotient (3) attains a maximum if and only if is the eigenvector of corresponding to the largest eigenvalue. Equivalently, the maximum is attained precisely when is the right singular vector corresponding to the largest singular value of , which throughout this paper will be denoted by . In this way we obtain a measure of finite-time predictability for a given initial condition .

In many applications it is often important to quantify the predictability of a certain event taking place in the future. We define an event to be a subset of the state space . For a given initial condition we say that the event occurs at time if , or, equivalently, . The predictability of the event can be quantified as follows. Assume that the dynamical system is equipped with an invariant probability measure supported on some attractor (in which case we also assume that ). This means that and for all measurable subsets . Then the distribution function of the time- singular values is given byThe conditional distribution of time- singular values given that the event occurs at time reads aswhere we have used that is an invariant measure so that . The predictability of the event can be quantified by comparing both distributions. For example, if the right endpoint of is much smaller than the right end point of , then the event can be called predictable. In the limit all events become equally predictable.

The advantage of the approach outlined in this section is the fact that it combines measures of predictability and the statistical recurrence properties of the system via its invariant measure. For simple dynamical systems for which the growth of errors can be computed analytically and for which the invariant measure is known the distributions (4) and (5) can be computed analytically. Hence, our approach may be used to derive general statements on the predictability of extremes for simple classes of dynamical systems in a rigorous way. This idea will be pursued in forthcoming work. Also note that the methodology applies to arbitrary events. This in particular includes the case of rare events, but these need not be extreme events in which some observable exceeds a threshold.

#### 3. Results

##### 3.1. The Lorenz-96 Model

In [35] Lorenz introduced a one-dimensional atmospheric model to study fundamental issues regarding the predictability of the atmosphere and weather forecasting. The model can be interpreted as a model for atmospheric waves traveling along a circle of constant latitude. We divide the latitude circle into equal sectors and define for the -th sector a distinct variable . The variables can be interpreted as meteorological quantities, such as pressure or vorticity, where the index of each variable plays the role of longitude. The dynamical equations arewith the periodic “boundary condition” . The dimension and forcing are free parameters. The Lorenz-96 model is often used to test data assimilation methods [42, 43] and subgrid scale parameterizations [44], for studies in statistical mechanics [45, 46], and in the general study of spatiotemporal chaos [47]. In this paper we use the Lorenz-96 model to study the predictability of extreme events in the vicinity of bifurcations.

The point is clearly an equilibrium solution of (6) for all and all . For all this equilibrium becomes unstable through either a supercritical Hopf or a double-Hopf bifurcation for [36]. In both cases a stable periodic attractor is born which has the physical interpretation of a traveling wave. Figure 1 shows the spatiotemporal properties of these waves: the period and the spatial wave number are plotted as a function of . In [36] it was proved analytically that the period tends to a finite limit as , but the wave number increases monotonically with .